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ACTIVE MATTER Active matter is a new field in non-equilibrium physics and describes systems whose elements are consuming energy and moving without external force. Example includes collections of cells and flocks of animals. Collective motion of a flock of birds. In this talk, however, we will focus on dynamics of a single non-biological particle as a first step to understand the mechanism of self-propulsion. single Dictyostelium(15 mm), Li et al, 2008.

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SELF-PROPELLED MOTION OF A NON-EQUILIBRIUM DROPLET How can we theoretically describe this spontaneous symmetry breaking under non- equilibrium condition? A. Shioi, T. Ban, and Y. Morimune, Entropy 12, 2308 (2010) K. Krischer and A. Mikhailov, Phys. Rev. Lett. 73, 3165 (1994). ) Self-propelled motion of a droplet with out-of-equilibrium concentration induced by Korteweg force (Binary mixture system). Bifurcation from a motionless state to a propagating state in a numerical simulation of reaction- diffusion equations Flow patterns around a self- propelling droplet. The blue line represents the direction of self- propelled motion.

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AIM OF OUR STUDY To construct a theoretical model of self- propelled motion induced by hydrodynamic effects under non-equilibrium environment. To derive the equation of the velocity of center of a droplet by analyzing the model. A Set of continuous field equations

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OUR MODEL SYSTEM We consider a phase separated binary mixture. We assume the existence of the third dilute component which influences the interfacial tension (a Marangoni effect). The third component is generated inside the droplet, diffuses away from the droplet and is decomposed due to chemical reactions. A rich B rich Yellow points represent the third Dilute component

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If there are concentration variations of the third dilute component on the surface of the droplet, the flow field will be generated around the droplet and the droplet will be transported by this flow field. MARANGONI EFFECT When we assume the profile as, the velocity field becomes Flow field around the droplet in the system which is moving with the droplet (We assume.)

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FORMULATION We consider phase separated binary mixture where a chemical reaction takes place inside a droplet. is a free energy density such that phase separation occurs. is composition of the dilute third component whose translational entropy is given by. The coefficient is assumed to depend linearly on. The density of the third dilute component The free energy density such that phase separation occurs.

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DYNAMIC EQUATIONS We neglect deformations of the droplet. The pressure is determined from the incompressibility condition. Diffusion constant Decomposition rate Generation rate Concentration of the dilute component

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OUTLINE OF OUR ANALYSIS ON THE DROPLET VELOCITY Concentration profile of the third dilute component on the interface Interface tension and its gradient Flow field around a droplet Marangoni effect The velocity of the droplet center We will obtain the expression of the droplet velocity in terms of the profile of the third dilute component. Interface

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THE VELOCITY OF THE CENTER OF A DROPLET We ignore the migration of the droplet due to the diffusion of. From geometrical consideration, the velocity of the center of a droplet is given as Normal component of the flow field on the surface of a droplet. Normal vector at the interface

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THE DROPLET VELOCOTY IN TERMS OF CONCENTRATION OF THE THIRD COMPONENT Here we can derive the following equation for the velocity of the center of a droplet by substituting the equation for in terms of the concentration profile. When we assume the profile as, we obtain the well-known results in hydrodynamics z

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ANALYSIS ON THE PROFILE OF THE DILUTE COMPONENT In terms of the Fourier transform, the compositional field satisfies. Here we dropped the convective term, assuming that it does not change the essential bifurcation behavior. We will later consider this point.

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SHORT TIME EXPANSION NEAR THE BIFURCATION THRESHOLD By assuming that the relaxation of the third component is sufficiently rapid compared with the motion of interface, we can solve the equation in the last slide in terms of the time derivative as We can evaluate the velocity of the center of mass of the droplet using this concentration profile of.

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DETERMINATION OF THE DROPLET VELOCITY The concentration profile of The droplet velocity The concentration profile of The droplet velocity and its time derivatives

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SCALED EQUATION OF MOTION FOR A DROPLET This equation indicates that there is a bifurcation from a motionless state to a uniformly moving state by changing the strength of the Marangoni effect!

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DISCUSSION The mechanism of self-propulsion Assume that a droplet was initially motionless, and is shifted slightly now. Then, the concentration of the third component becomes higher on the rear of the droplet, and the interface tension also becomes higher on the rear! Before After Shifted slightly

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The effect of convective term The correction for profile of to the first order is given as. This changes as. DISCUSSION However, this correction does not change the essential bifurcation behavior.

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CONCLUSION We have constructed a theoretical model of self-propelled motion induced by hydrodynamic effects under non-equilibrium environment. We have derived the equation of the velocity of center of mass of a droplet, and found that it indicates a bifurcation from a motionless state to a uniformly propagating state.

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PROFILE OF THE THIRD COMPONENT By performing inverse Fourier transformation, we obtain the profile of the third component as where

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By substituing of the profile of into the equation of the velocity of the center of the mass, we obtain the closed equaition for the velocity of the center of the mass of a droplet, where the coefficients are given as EQUATION OF MOTION FOR A DROPLET.